I noticed that layering Henon attractor images with avalue=1 and bvalues from -0.2 to 0.3 looks like a distorted version of the logistic map.
In the image below you can see the layered images (left).
After that I tried to accentuate the first "branches" (right)

Very nice! Have you read the book Chaos: Making new Science by James Gleick? It covers dynamical systems, including Henon attractor and logistic map. I remember it explained a lot, maybe you would find your answer there and, while looking for it, you would enjoy reading through the rest.
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David ČepelíkApr 24 '13 at 11:40

"Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. It was discovered in 1978.[1]"